| Don Bowey 2007-07-04, 12:33 pm |
| On 7/4/07 7:52 AM, in article 468bb3c0$0$24780$4c3
68faf@roadrunner.com, "Ron
Baker, Pluralitas!" <this@aint.me> wrote:
>
> "Keith Dysart" <Keith.Dysart@gmail.com> wrote in message
> news:1183489552.088116.51340@g4g2000hsf.googlegroups.com...
>
> You win. :)
>
> When I conceived the problem I was thinking
> cosines actually. In which case there are no
> phase shifts to worry about in the result.
>
> I also forgot the half amplitude factor.
>
> While it might not be obvious, the two cases I
> described are basically identical. And this
> situation occurs in real life, i.e. in radio signals,
> oceanography, and guitar tuning.
>
> It follows from what is taught in high school
> geometry.
>
> cos(a) * cos(b) = 0.5 * (cos[a+b] + cos[a-b])
>
> Basically: multiplying two sine waves is
> the same as adding the (half amplitude)
> sum and difference frequencies.
No, they aren't the same at all, they only appear to be the same before
they are examined. The two sidebands will not have the correct phase
relationship.
One could, temporarily, mistake the added combination for a full carrier
with independent sidebands, however.
>
> (For sines it is
> sin(a) * sin(b) = 0.5 * (cos[a-b]-cos[a+b])
> = 0.5 * (sin[a-b+90degrees] - sin[a+b+90degrees])
> = 0.5 * (sin[a-b+90degrees] + sin[a+b-90degrees])
> )
>
> --
> rb
>
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